Folia Mathematica Acta Universitatis Lodziensis Vol. 55–58, No. , pp. TOPOLOGIES OF HASHIMOTO TYPE WITH RESPECT TO SIGMA-IDEAL OF COUNTABLE SETS SEBASTIAN LINDNER‡ c 2004 for University ofL´od´zPress Abstract. The paper deals with properties of and interrelations between three topologies on the real line finer than the natural topology. Also classes of real functions of a real variable, which are continuous, when the domain and the range are equipped with one of these topologies, are examined. Topologies are similar to that studied in [3] and some properties of classes of continuous functions resemble that in [1]. In the sequel will stand for a natural topology on the real line R, J0-for the σ-ideal of countableT subsets of R, and Q-for the set of rational numbers. Proposition 1. (See [3]). The family of sets = G R : Go∈T I∈J0 G = G I T1 ⊂ ∃ ∃ 0 \ is a topology finer than the natural topology. Let As s∈S = R/ , where is the equivalence relation defined by x y iff{ x} y Q. Recall∼ that card(∼ S) = continuum. ∼ − ∈ Proposition 2. The family of sets ∞ = B R : B = (a, b) [ Asn for some a, b R B { ⊂ \ ∈ n=1 and some sequence sn n∈N of elements of S { } } is a basis for a topology on the real line. Proposition 3. The family of sets n = C R : C = (a, b) [ Asi for some a, b R C { ⊂ \ ∈ i=1 n and some finite sequence si of elements of S { }1 } ‡Faculty of Mathematics, University ofL´od´z , Banacha 22 St., 90–238L´od´z, Poland. E-mail: [email protected]. Key words and phrases: Hashimoto topology. AMS subject classifications: 54A10. 55 56 SEBASTIAN LINDNER is a basis for a topology on the real line. The proofs of Propositions 2 and 3 are straightforward. Observe that for each B ,B (C ,C , respectively) if B B = (C C = , 1 2 ∈ B 1 2 ∈ C 1 ∩ 2 6 ∅ 1 ∩ 2 6 ∅ respectively), then B1 B2 (C1 C2 , resp.). For properties of a basis see, for example [2].∩ ∈ B ∩ ∈ C In the sequel we shall denote by 2 ( 3, resp.) a topology generated by ( , resp.). Obviously and areT finerT than a natural topology. B C T2 T3 Proposition 4. 6 6 6 . T ⊂= T3 ⊂= T2 ⊂= T1 Proof. The inclusions are obvious. To prove the inequalities observe, that if G = ( 1, 1) [Q 0 ], − \ \ { } then G . Indeed, G by definition. Suppose now, that G . ∈T1 \T2 ∈T1 ∈T2 Then G is an union of elements of , G = Bt for some T , where B St∈T Bt for t T . Consequently, there exists t0 T such that 0 Bt0 . ∈ B ∈ ∞ ∈ ∈ Since Bt0 = (a , b ) Asi for some a , b R and si i∈N from S, 0 0 \ Si=1 0 0 ∈ { } then Bt0 contains all rational numbers from the interval (a0, b0). But this contradicts the inclusion Bt0 G and G does not contain any rational number different from 0 . ⊂ ∞ Suppose now that B = (0, 1) Si=1 Asi , where si = sj for i = j. It is not difficult to see that B .\ 6 6 ∈T2 \T3 Proposition 5. Ti is Hausdorff and not regular for i 1, 2, 3 . ∈ { } Proof. All considered topologies are Hausdorff, because they are finer than the natural topology. To prove the second assertion take F = [1, 2] Q and ∩ x = √2. Obviously F is closed in i (for i 1, 2, 3 ). 0 T ∈ { } Suppose that there exist U , U such that F U , √2 U , U 1 2 ∈ T1 ⊂ 1 ∈ 2 1 ∩ U2 = . Hence U2 = G2 I2, where G2 is open in the natural topology and I is a∅ countable set. Let−q [1, 2] Q G . Then there exists G -open in 2 ∈ ∩ ∩ 2 1 the natural topology and a countable set I1 such that q G1 I1 U1. Therefore q (G G ), so (G G )-nonempty open set.∈ Then− ⊂ ∈ 2 ∩ 1 2 ∩ 1 (G I ) (G I ) = (G G ) (I I ) = , 2 − 2 ∩ 1 − 1 2 ∩ 1 \ 2 ∪ 1 6 ∅ so U1 U2 = is a contradiction. Also∩F and6 ∅x cannot be separated in coarser topologies and . 0 T2 T3 Let C( , i) be a class of all continuous functions f : (R, ) (R, i) for i 1,T2,T3 . T → T ∈ { } Proposition 6. C( , ) is a class of all constant functions. T T3 TOPOLOGIES OF HASHIMOTO TYPE 57 Proof. Suppose that there exists a function f C( , 3), which is not con- stant. So f(x ) < f(x ) for some x = x .∈ ForT instanceT suppose that 1 2 1 6 2 x1 < x2. Let G be the set of all irrational numbers. Then G 3 and −1 −1 ∈ T f (G) . Hence f (G) = (an, bn), where the union is at most ∈ T Sn countable. Since 3, f C( , ). If f were constant on each −T1 ⊂T ∈ T T (an, bn), then f(f (G)) would be at most countable, but it is impossi- ble, because f(f −1(G)) [f(x ),f(x )] Q. Then there exists n such ⊃ 1 2 \ 0 that f((an0 , bn0 )) is a non-degenerate interval, which is impossible since −1 f((an0 , bn0 )) f(f (G)) G. This contradiction ends the proof. ⊂ ⊂ Corollary 1. C( , ) and C( , ) consist only of constant functions. T T1 T T2 Proof. The proof is obvious by virtue of Proposition 4. From Theorem 34.1 in [4], p. 78 it follows immediately, that Proposition 7. C( , )= C( , ). T1 T T T Corollary 2. C( , )= C( , )= C( , ). T2 T T3 T T T Proposition 8. The topological spaces (R, 1) and (R, 2) are not homeo- morphic. T T Proof. Suppose that f : (R, 2) (R, 1) is a homeomorphism. Then f is bijective and ordinary continuousT → function,T by virtue of Corollary 2, since f C( 2, ). Take G = ( 1, 1) [Q 0 ]. Then G 2, by virtue of proof∈ ofT PropositionT 4. But−f(G)=\ f((\1 {, 1))} f([Q 6∈0 ]) T is a difference − \ \ { } of a non-degenerate open interval and a countable set, so f(G) 1-a contradiction. ∈ T Proposition 9. The spaces (R, 2) and (R, 3) are not homeomorphic. Moreover, the class C( , ) is equalT to the classT of all constant functions. T3 T2 Proof. Let f C( 3, 2). Let s1,s2,... be a sequence of different ele- ments of a set∈S, describedT T in the{ connection} with R/ . Observe, that: ∞ ∼ (1) Si=1 Asi is not closed in 3, moreover, the closure of this set (in 3) is equal to R. T T ∞ ∗ (2) The set F = Si=1 f(Asi ) is at most countable. Then F = Sx∈F [x]∼ ∞ −1 ∗ −1 ∗ is closed in 2. From the facts that Si=1 Asi f (F ), and f (F ) is T −1 ∗ ⊂ closed in 3, and (1) we have f (F )= R. To sumT up: f C( , ), and f(R) is countable. Then f is constant. ∈ T T References [1] V. Aversa, W. Wilczy´nski, Classes of continuous functions, Atti Sem. Mat. Fis. Univ. Modena 44 (1996), pp. 385–389 [2] R. Engelking, General Topology, Heldermann, Berlin 1989. 58 SEBASTIAN LINDNER [3] H. Hashimoto, On the ∗ topology and its applications, Fund. Math., 91 (1976), pp. 5–10. [4] B. Thomson, Real functions, Lecture Notes in Mathematics, Springer–Verlag Berlin 1985..